Symmetric Teleparallel and Nonmetricity Gravity

Symmetric teleparallel gravity represents general relativity with an affine connection that is flat and torsion-free but not metric compatible. In a local coincident gauge the connection coefficients vanish, so nonmetricity carries the gravitational information. A particular quadratic nonmetricity scalar differs from the Levi-Civita Ricci scalar by a divergence; its nonlinear replacements, such as $f(\mathcal Q)$, are new theories rather than alternative notation for general relativity.

Required background. Frames, Tetrads, and Spinors supplies frame changes; Differential Forms and Integration supplies curvature and torsion; and Gravitational Actions and Boundary Terms supplies the variational role of the divergence.

Helpful background. Einstein Equations gives the equivalent metric dynamics, while Formulations of General Relativity provides a broader comparison of variables and gauges.

Flat, torsion-free connection and coincident gauge

Let $g_{\mu\nu}$ and $\Gamma^\rho{}_{\mu\nu}$ be independent geometric data. Symmetric teleparallel geometry imposes

$$R^\rho{}_{\sigma\mu\nu}(\Gamma)=0, \qquad T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}=0,$$

while allowing $Q_{\alpha\mu\nu}=-\nabla_\alpha g_{\mu\nu}$ to be nonzero. Flatness and vanishing torsion imply locally

$$\Gamma^\alpha{}_{\mu\nu} =\frac{\partial x^\alpha}{\partial\xi^\lambda} \partial_\mu\partial_\nu\xi^\lambda$$

for four functions $\xi^\lambda(x)$ with invertible Jacobian. Choosing the $\xi^\lambda$ as coordinates gives the coincident gauge $\Gamma^\alpha{}_{\mu\nu}=0$. This is a coordinate gauge, not a tensor assertion: in another coordinate system the same connection generally has nonzero coefficients. Globally, nontrivial topology or affine holonomy can obstruct one coincident chart covering the whole spacetime, although local patches remain available. The pure-gauge form and its Stückelberg interpretation are derived in Beltrán Jiménez, Heisenberg, and Koivisto 2018, §II, pp. 2–3, Eqs. (17)–(23), PDF.

The site nonmetricity scalar

Define the two traces

$$Q_\alpha=Q_{\alpha\mu}{}^\mu, \qquad \widetilde Q_\alpha=Q^\mu{}_{\alpha\mu}.$$

For the site signature and $Q_{\alpha\mu\nu}=-\nabla_\alpha g_{\mu\nu}$ convention, the required quadratic combination is

$$\mathcal Q =\frac14Q_{\alpha\mu\nu}Q^{\alpha\mu\nu} -\frac12Q_{\alpha\mu\nu}Q^{\mu\alpha\nu} -\frac14Q_\alpha Q^\alpha +\frac12Q_\alpha\widetilde Q^\alpha.$$

The signs in this definition, its boundary identity, and the action below form one convention package. The original coincident-GR paper defines $Q^{\mathrm{src}}_{\alpha\mu\nu}=+\nabla_\alpha g_{\mu\nu}=-Q_{\alpha\mu\nu}$ and chooses the opposite quadratic scalar, $Q_{\mathrm{src}}=-\mathcal Q$. These are two separate choices: reversing the tensor alone leaves every quadratic contraction unchanged. The linear trace vector in the boundary identity does reverse. Keeping these translations separate avoids attributing a scalar sign change to the wrong operation.

Because the connection is torsion-free, its distortion relative to Levi-Civita is the disformation

$$\Gamma^\rho{}_{\mu\nu} =\mathring\Gamma^\rho{}_{\mu\nu}+L^\rho{}_{\mu\nu}, \qquad L^\rho{}_{\mu\nu} =\frac12\left( Q_\mu{}^\rho{}_{\nu} +Q_\nu{}^\rho{}_{\mu} -Q^\rho{}_{\mu\nu} \right).$$

This formula is also a sign check: substituting it into $-\nabla_\alpha g_{\mu\nu}$ returns $Q_{\alpha\mu\nu}$.

Curvature–nonmetricity identity and action

Insert $\Gamma=\mathring\Gamma+L$ into the curvature-difference formula and set $R(\Gamma)=0$. Contract the first and third curvature indices and then with the metric. The quadratic disformation terms give $\mathcal Q$, while the derivatives reduce to the two trace vectors. The result is the off-shell identity

$$\boxed{ \mathring R =-\mathcal Q +\mathring\nabla_\mu \left(Q^\mu-\widetilde Q^\mu\right) }.$$

Equivalently, the last term is $(\sqrt{-g})^{-1}\partial_\mu[\sqrt{-g}(Q^\mu-\widetilde Q^\mu)]$. Hence

$$S_{\mathrm{STEGR}} =-\frac{1}{2\kappa} \int_{\mathcal M}d^4x\,\sqrt{-g}\,\mathcal Q, \qquad \kappa=8\pi G,$$

and the Einstein–Hilbert action have the same bulk metric equation when their matter couplings, boundary data, and boundary terms are matched. The connection equation is then redundant with the diffeomorphism identity under the standard metric matter coupling. The general distortion derivation and boundary qualification appear in Bahamonde et al. 2023, §2.3, pp. 17–19, Eqs. (2.45)–(2.53), PDF; their signature-dependent quantities have been translated together here.

Independent flat-FLRW check

Use the coincident gauge and Cartesian flat-FLRW metric

$$ds^2=-dt^2+a^2(t)\delta_{ij}dx^i dx^j, \qquad H=\frac{\dot a}{a}.$$

The only nonzero components are

$$Q_{0ij}=-2a^2H\delta_{ij}, \qquad Q_0=-6H, \qquad \widetilde Q_\alpha=0.$$

The four terms in $\mathcal Q$ can now be checked separately:

$$\frac14Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}=-3H^2, \qquad -\frac12Q_{\alpha\mu\nu}Q^{\mu\alpha\nu}=0,$$ $$-\frac14Q_\alpha Q^\alpha=9H^2, \qquad \frac12Q_\alpha\widetilde Q^\alpha=0.$$

Therefore

$$\boxed{\mathcal Q=6H^2}.$$

Moreover, $Q^0=6H$ and $\widetilde Q^0=0$, so

$$\mathring\nabla_\mu(Q^\mu-\widetilde Q^\mu) =\frac{1}{a^3}\frac{d}{dt}(6a^3H) =6\dot H+18H^2.$$

The identity gives $\mathring R=-6H^2+6\dot H+18H^2=6(\dot H+2H^2)$, exactly the Levi-Civita result. This independently fixes the scalar and boundary signs. The corresponding primary calculation states the coincident-gauge FLRW result in Beltrán Jiménez, Heisenberg, and Koivisto 2018, §III, pp. 3–4, Eqs. (24)–(27), PDF.

Modified nonmetricity dynamics and gauge limits

Replacing $\mathcal Q$ by $f(\mathcal Q)$ breaks the boundary-term equivalence for $f''\ne0$. The connection remains constrained to be flat and torsion-free, but its equation is no longer merely the STEGR boundary identity in disguise. One should vary the covariant constrained action—or its Stückelberg form—and choose coincident gauge afterward. Setting $\Gamma=0$ before variation without tracking the lost connection variation can erase an equation or invite an incompatible second coordinate condition.

The coincident gauge also does not mean that gravitation has vanished. In that gauge $Q_{\alpha\mu\nu}=-\partial_\alpha g_{\mu\nu}$, so the metric derivatives carry the dynamics. Conversely, a coordinate transformation that produces nonzero $\Gamma$ has not created a new force. The invariant comparison uses the complete metric–connection pair.

For $f(\mathcal Q)=\mathcal Q+2\Lambda$ in the site action with its overall minus sign, the GR limit is recovered. For a nonlinear function, branches, perturbations, and matter coupling require their own constraint and stability analysis. Equality of a homogeneous Friedmann equation is not proof of full equivalence; perturbations can distinguish models with the same background expansion.

Empirical likelihoods and numerical bounds are beyond this derivation and belong to Tests of Gravity. Claims about a stable low-energy quantum-gravity EFT are also separate: rewriting the classical gravitational action in nonmetricity variables does not supply a quantum completion.

Common pitfalls

Treating coincident gauge as an extra field equation. It is a local coordinate choice permitted by flatness and zero torsion. It does not hold as a coordinate-independent statement.

Mixing nonmetricity conventions. Reversing $Q_{\alpha\mu\nu}$ reverses its traces and their divergence, but does not reverse a quadratic scalar with its coefficients held fixed. Some authors independently define the negative scalar and compensate with the opposite action sign. Check both definitions before using a quoted FLRW value.

Exercises

Consider a de Sitter patch in coincident Cartesian FLRW coordinates, $a(t)=e^{H_0t}$ with constant $H_0$. Verify the curvature–nonmetricity identity without computing any Christoffel symbol explicitly.

Solution

The component contractions above give $\mathcal Q=6H_0^2$. The only boundary vector component is $Q^0-\widetilde Q^0=6H_0$, so its Levi-Civita divergence is $a^{-3}d(6a^3H_0)/dt=18H_0^2$. Therefore $-\mathcal Q+\mathring\nabla_\mu(Q^\mu-\widetilde Q^\mu)=12H_0^2$, which is the Ricci scalar of four-dimensional de Sitter space in the site curvature convention.

References

• Bahamonde, S., et al., “Teleparallel Gravity: From Theory to Cosmology,” Reports on Progress in Physics 86 (2023) 026901. DOI. Open PDF
• Beltrán Jiménez, J., L. Heisenberg, and T. Koivisto, “Coincident General Relativity,” Physical Review D 98 (2018) 044048. DOI. Open PDF